Enrollments at 2- and 4-year colleges have remained flat, while the number
of AP Calculus tests has continued to grow at about 6% per year.[4]

Of course, these are only partial numbers. First-semester calculus is taught
in other terms. But at most colleges, well over half of the enrollments in
the first calculus class are in the fall term. It is safe to say that total
college enrollment in mainstream first-term calculus is under 500,000, probably
closer to 400,000. On the other hand, not all or even most students who study
calculus in high school take the AP exam. According to the latest NAEP study,
16% of 12th-graders in 2004–05 were taking or had taken a course called
Calculus.[5] This implies that about 500,000 high school
students studied calculus in 2004–05.[6]

The problem is that relatively few of these students earn recognition for
college-level work in calculus. Most colleges give credit beginning at either
a 3 or a 4 on the AP Calculus exams. In 2006, 173,000 of the test takers earned
3 or higher, 129,000 earned 4 or higher.[7] Students do
earn college credit for calculus in other ways including the International
Baccalaureate program, enrollment in local colleges, and dual-enrollment programs
(about which I will say much more in a future column), but these numbers are
relatively small, a few tens of thousands. The total number of students who
entered college in 2006 and actually received college credit for calculus
taken while in high school was almost certainly below 200,000, roughly a third
of those who had studied calculus.

What is worse, many of the students who arrive in college with a calculus
course on their high school transcript are not even considered ready for calculus.
College-administered placement exams often direct them to pre-calculus or
college algebra. I am unaware of any estimate of the number of such students,
but anecdotally it appears to be high.
This constitutes a crisis. Students who choose to study calculus in high school
are a reasonable approximation of the top quintile of high school graduates.
They are some of our best candidates for demanding careers in science, technology,
engineering, and the mathematical sciences. I do not believe that, on the
whole, we who teach in the colleges serve any of them very well. Those who
have received college credit for the calculus they studied often find themselves
in courses that articulate poorly with their high school experience. Those
who start college by repeating first-term calculus find themselves re-treading
familiar territory, but at a much faster pace, in larger classes, and with
an instructor who is unable to give them the individual attention that they
experienced when they struggled with these ideas the previous year. Those
who are directed back to pre-calculus or college algebra find themselves in
a frustrating and humiliating position with regard to mathematics.

The MAA and NCTM are very much aware of this problem. Bernie Madison, as
chair of the MAA Committee on Articulation and Placement, has made it one
of his top priorities. The recently constituted MAA-NCTM Committee on Mutual
Concerns, chaired by Ann Watkins, has chosen as its first item of focus the
problems of articulating calculus instruction in the high schools with that
in colleges. There are three urgent tasks that we face at this time.

First, we need a much better grasp of the extent and nature of the problem.
Of the students who receive college credit for calculus studied in high school,
how many never go on to the next mathematics class? How many choose to retake
the class they are entitled to skip? How many continue but encounter insurmountable
difficulties making the transition to college classes? What are the most successful
strategies for these students and how do these choices affect their perception
of mathematics and ability to complete the mathematical training needed for
their chosen careers? Of the students who retake the course they studied in
high school, how many of them now succeed? How many of them, in retrospect,
feel that they have made good use of their time? What are the most successful
strategies for these students and how do these choices affect their perception
of mathematics and ability to complete the mathematical training needed for
their chosen careers? How many students take calculus in high school but are
deemed inadequately prepared to study calculus when they get to college? What
are the most successful strategies for these students and how do these choices
affect their perception of mathematics and ability to complete the mathematical
training needed for their chosen careers?

Second, we need to communicate our findings to high school students, their
parents, their teachers and counselors. We need a clear set of expectations
of the preparation needed before beginning the study of calculus. We need
collaborative ventures within the mathematical community, embracing both high
school and college teachers, that articulate and disseminate these expectations
and assist K-12 mathematics teachers in preparing students to meet these expectations.
Several national organizations have been working on such sets of expectations,
especially NCTM[8], The College Board[9],
and Achieve[10] (an organization that works
with the National Governor’s Association). The MAA’s PMET (Preparing
Mathematicians to Educate Teachers) has begun this work, and the MAA special
interest group SIGMAA TAHSM (Teaching Advanced High School Mathematics) is
dedicated to fostering the dialogue between college and high school teachers,
but there is much left to do.

Third, we need to use these findings to re-evaluate the way we teach calculus
in college. Most colleges still teach a calculus sequence predicated on the
assumption that students taking the first course in the sequence have never
seen calculus before, and students in subsequent courses have come through
the previous courses at that college. The norm today is that most students
taking first-term calculus have studied calculus in high school, and most
of them have no intention of continuing the study of mathematics beyond that
course. The norm in most colleges is that a significant number, often a majority,
of the students taking the second course in calculus has earned credit for
calculus learned in high school. This is their first mathematics course in
college. Once we know what is working and what is not, where we are failing
to meet the needs of our students, then we can revise our curricula to meet
these needs.

The landscape has changed dramatically in just the past few years. There
is a juggernaut pushing calculus into the high schools, a massive and inexorable
force driven by competition to get into college, to get into the best colleges,
to get into the best colleges with generous financial aid. Whether or not
we believe that calculus should be studied in high school, we cannot afford
to ignore the fact that it has moved into the high school curriculum and is
increasingly seen as a prerequisite for college. When and how we teach calculus
carries implications all the way back to pre-Kindergarten and all the way
forward into graduate school. We need an accurate assessment of the strains
that the movement of calculus into the high school curriculum is putting on
our educational system and an identification of the points of fracture. Armed
with such an assessment, we will be prepared for the intensive and sustained
effort that will be required as we address the articulation of high school
to college mathematics and its repercussions throughout the entire mathematics
curriculum.

[8] While not a set of standards, the recent NCTM Focal
Points for Prekindergarten through Grade 8 Mathematics does articulate basic
expectations for mathematics education for pre-K through grade 8. See www.nctm.org/focalpoints/

We would appreciate more examples that document experiences with the use of
technology as well as examples of interdisciplinary cooperation.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester
College in St. Paul, Minnesota, he was one of the writers for the Curriculum
Guide, and he currently serves as Chair of the CUPM. He wrote this column
with help from his colleagues in CUPM, but it does not reflect an official
position of the committee. You can reach him at bressoud@macalester.edu.